Can anyone explain why the codomain fibration cod: C^\rightarrow -> C, which requires pull-backs, gets loads of attention, while the domain fibration dom: C^rightarrow -> C, which works for all C, hardly gets a look in? Is the dom fibration really such a poor relation?
I have a couple of points to make about this. 1) By duality, the "poor relation" fibration dom can be viewed as the opfibration cod. So it is not so poor after all, just part of an even richer structure of cod. 2) Fibrations over C "amount" to pseudofunctors C^op --> Cat. Let S be the pseudofunctor S(u) = C/u on objects, using pullback on arrows; this is the "rich guy". The "poor guy" T is a covariant functor (not just pseudo) T(u) = C/u , using composition on arrows. To get a contravariant Cat-valued functor on C, follow T by the presheaf construction P : Cat^coop --> Cat. It turns out then that the Yoneda embedding gives a fully faithful pseudonatural transformation y : S --> PT. Now PT is a very important character; every internal full subcategory of the topos E of presheaves on C is a full subobject of PT. (A good example is where E is globular sets and PT is the globular category of higher spans.) It is true that S can be thought of as an internal model of E in Cat(E) leading to indexed (or parametrized) category theory. But the reason this works well is that S is a full subobject of PT. Again T wins out! Regards, Ross